The involution \(\widetilde{\iota }\)  defines a \(\mathbb {Z}/2\mathbb {Z}\)  action on \(\widetilde{C}\)  whose quotient is \(\widetilde{D}\) . Since \(p_1, p_2\)  are in the fixed locus of the involution \(\widetilde{\iota }\)  on \(\widetilde{C}\) , the line bundle \(\mathcal {O}_{\widetilde{C}}(p_1-p_2)\)  is \(\mathbb {Z}/2\mathbb {Z}\)  invariant and hence, is \(\mathbb {Z}/2\mathbb {Z}\) -linearizable (cf. [1]}). Moreover, there are only two possible \(\mathbb {Z}/2\mathbb {Z}\)  linearizations. The Kempf Descent Lemma (see [2]}) says that \(\mathcal {O}_{\widetilde{C}}(p_1-p_2)\)  descends to a line bundle on \(\widetilde{D}\)  if and only if there exists a \(\mathbb {Z}/2\mathbb {Z}\) -linearization of \(\mathcal {O}_{\widetilde{C}}(p_1-p_2)\)  such that the linearized action on the fibers of \(\mathcal {O}_{\widetilde{C}}(p_1-p_2)\)  over the fixed points \(p_1,p_2,q_1,q_2\)  of \(\widetilde{\iota }\)  is trivial. Considering \(\mathcal {O}_{\widetilde{C}}(p_1-p_2)\)  as the subsheaf of the sheaf of rational functions consisting of rational functions on \(\widetilde{C}\)  vanishing at \(p_2\)  and having at most a simple pole at \(p_1\) , one can use local coordinates around \(p_1,p_2,q_1,q_2\)  in which the \(\mathbb {Z}/2\mathbb {Z}\) -action on \(\widetilde{C}\)  is linear to check that there is no \(\mathbb {Z}/2\mathbb {Z}\) -linearization satisfying the conditions of Kempf's Descent Lemma.

